# Independent double Roman domination in graphs

## TL;DR

This paper introduces the concept of independent double Roman domination in graphs, explores its properties and bounds, and characterizes possible pairs of domination numbers in trees.

## Contribution

It defines the independent double Roman domination number, establishes bounds relating it to other domination parameters, and characterizes realizable pairs in trees.

## Key findings

- Established bounds on IDRDN in terms of graph order, degree, and edge cover
- Connected IDRDN with independent domination and Roman domination numbers
- Characterized all feasible pairs of IDN and IDRDN in non-trivial trees

## Abstract

An independent double Roman dominating function (IDRDF) on a graph $G=(V,E)$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ has at least two neighbors assigned $2$ under $f$ or one neighbor $w$ with assigned $3$ under $f$, and if $f(v)=1$, then there exists $w\in N(v)$ with $f(w)\geq2$ such that the positive weight vertices are independent. The weight of an IDRDF is the value $\sum_{u\in V}f(u)$. The independent double Roman domination number $i_{dR}(G)$ of a graph $G$ is the minimum weight of an IDRDF on G. We initiate the study of the independent double Roman domination and show its relationships to both independent domination number (IDN) and independent Roman $\{2\}$-domination number (IR2DN). We present several sharp bounds on the IDRDN of a graph $G$ in terms of the order of $G$, maximum degree and the minimum size of edge cover. Finally, we show that, any ordered pair $(a,b)$ is realizable as the IDN and IDRDN of some non-trivial tree if and only if $2a + 1 \le b \le 3a$.